My class was canceled due to the coronavirus and I am supposed to solve the following problem:

Assume that the consumer consumes only two goods, and his utility function is $u=x_{1}x_{2}$. Last year the prices of both goods were 10 and the consumer income was 100. This year the price of the first good is still 10, but the price of the second rose to 25. Calculate Paasche and Laspeyres price indices.

I know how to calculate them using formulas if I have quantity as well as price of products.

But how should I calculate it, if I have income and utility function?


  • $\begingroup$ Hint: as a first step find the quantities and prices by solving the consumer's optimization problem :) $\endgroup$
    – noob2
    Mar 20 '20 at 13:18
  • $\begingroup$ @noob2 I was trying to calculate it using utility maximization, I was doing it with the help of youtube video and I got that $x_{1}=x_{2}$, is that correct? $\endgroup$
    – Peter
    Mar 20 '20 at 13:41
  • $\begingroup$ Yeah, u(.,.) is symmetric and in the first period the goods have the same price so they will be consumed in equal amounts (x1=x2=5 (check my math)). However in the second period the second good becomes more expensive so the consumer will buy less of it and more of the first, so you will find two different numbers for $x_1,x_2$. Keep going. $\endgroup$
    – noob2
    Mar 20 '20 at 14:32
  • $\begingroup$ @noob2 then in the second period, $x_{1}= 5 $ and $x_{2}=2.5?$ $\endgroup$
    – Peter
    Mar 20 '20 at 14:41
  • $\begingroup$ How could that be? The budget is 100 and in your "solution" he would be spending $5*10+2.5*25 = 112.5$ i.e more money than he has available. The budget constraint $p_1 x_1 + p_2 x_2 = 100$ needs to be satisfied. $\endgroup$
    – noob2
    Mar 20 '20 at 14:48

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